Rotate Points Around an Arbitrary Axis
(Vertical Axis Method)

Rotations around an arbitrary axis are about the most complex stereonet
constructions. They arise particularly when data from folded or tilted rocks
must be restored to their original orientations when the rocks were horizontal.
Current directions and magnetic pole orientations are examples. Multiple
rotations also arise in universal-stage research in petrology.

The general sequence of steps necessary is as follows:

Plot the data and the rotation axis.

Rotate the rotation axis to a location where the data can be rotated on a
stereonet. This means rotating the axis to either a vertical or horizontal
orientation. In this example we will rotate the axis to a vertical position.
This example and the horizontal-axis example
use the same data.

Rotate the data around the rotated axis.

Rotate the rotated data and axis so that the rotated axis returns
to its original orientation.

In other words everything has to be rotated at least twice, not counting
rotations of the overlay. For very large data sets it may just pay to construct
an oblique stereonet. (In my undergraduate structural geology class we were
given a rotation problem involving 300 points! For something like that, you
pretty much need to do each stage of the problem on a separate overlay.)

Example

Given a line trending 318 and plunging 37 degrees, rotate the line 70 degrees
around an axis trending 025 and plunging 60 degrees.

1. Plot the line (red) and the rotation axis (blue).

2. We need to get the axis down to the equator to rotate it to a
vertical position. Rotate the overlay 65 degrees (90-25). The initial
position of the point and axis are shown as open squares, the final
positions as solid.

3. Rotate the axis to a vertical position. Since it already plunges 60
degrees, another 30 degree rotation is needed. Rotate the line the same
amount.

4. Now rotate the line by 70 degrees around the rotation axis.

5. Now we undo the previous rotations. Rotate the axis back
to its original plunge of 60 degrees, and rotate the rotated line
the same amount. This reverses the rotation in step 3. From here on the
original position of the line is shown in green.

6. Now rotate the axis back to its original trend of 025 degrees,
reversing the rotation in step 2. Rotate the rotated line as well.

7. The final result. The original line is shown as an open square, the
rotated line as a solid square. The rotated position trends 008 degrees
and plunges 14 degrees.

8. As a check, the angle between the axis and both positions of the
line should be the same (47 degrees in this case).

The original and rotated positions of the line should both lie on a small
circle centered on the rotation axis (shown in purple in figure 8). Note that
this construction and the horizontal-axis method give the same result, as they
should.